The 192.168.0.1 is an IPv4 dot-decimal notation. The network address.
The /24 is the subnet mask as a host identifier (to denote its own network).This is a neat explanation.

This is the mainline of how the tool above does the calculation. Steps:

Convert the IPv4 dot-decimal notation to dot-binary notation.

192.168.0.1 ► 11000000.10101000.00000000.00000001

Translate the subnet mask suffix, in above example: /24, into dot-binary notation.The total length of the notation is 32 bits (32 of 0's and/or 1's).Look at the number after the slash, it is 24. Meaning we create 24 of 1's from the left (24 leading 1-bits). The last 8 bits (32 - 24 = 8) are filled 0's.Now we have the binary version of the suffix.Put dot (.) to separate each segment with 8 bits interval (dot-binary notation).

Or, you could invert the subnet mask binary version, then convert it back to decimal.11111111.11111111.11111111.00000000 inverted becomes 00000000.00000000.00000000.1111111100000000.00000000.00000000.11111111 to decimal is 0.0.0.255.Same result.

The last IP then is calculated by adding the subtraction result number 6 above with the first IP we got from step number 4.

COMPLETE.So then 192.168.0.1/24 is to tell the computer that its network HOST is at 192.168.0.0 (and the network block is within 192.168.0.0 — 192.168.0.255).

That's the main idea for translating the /number suffix (network block) in IPv4 CIDR notation. It's not limited to this method, there are others, but the principles are like those above.

For more information about the subnet mask CLASS, IP TYPE, usable addresses within a block, preassigned addresses and things related to that, you could browse networking forums / websites / articles on Wikipedia.